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We develop a framework that allows the use of the multi-level Monte Carlo(MLMC) methodology (Giles2015) to calculate expectations with respect to theinvariant measure of an ergodic SDE. In that context, we study the(over-damped) Langevin equations with a strongly concave potential. We showthat, when appropriate contracting couplings for the numerical integrators areavailable, one can obtain a uniform in time estimate of the MLMC variance incontrast to the majority of the results in the MLMC literature. As aconsequence, a root mean square error of $\mathcal{O}(\varepsilon)$ is achievedwith $\mathcal{O}(\varepsilon^{-2})$ complexity on par with Markov Chain MonteCarlo (MCMC) methods, which however can be computationally intensive whenapplied to large data sets. Finally, we present a multi-level version of therecently introduced Stochastic Gradient Langevin Dynamics (SGLD) method(Welling and Teh, 2011) built for large datasets applications. We show thatthis is the first stochastic gradient MCMC method with complexity$\mathcal{O}(\varepsilon^{-2}|\log {\varepsilon}|^{3})$, in contrast to thecomplexity $\mathcal{O}(\varepsilon^{-3})$ of currently available methods.Numerical experiments confirm our theoretical findings.

Multilevel Monte Carlo methods for the approximation of invariant measures of stochastic differential equations